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%I #21 Jan 20 2024 18:41:20
%S 1,2,5,10,20,34,59,94,149,224,334,480,685,950,1307,1762,2357,3100,
%T 4050,5220,6685,8466,10659,13294,16494,20298,24859,30234,36609,44056,
%U 52806,62952,74770,88380,104112,122116,142786,166304,193134,223504,257954,296756,340544
%N Expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2.
%C Molien series for S_4 X S_4, cf. A001400.
%H Colin Barker, <a href="/A117486/b117486.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).
%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2.
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) - 4*a(n-5) + 4*a(n-6) + 4*a(n-7) + 2*a(n-8) - 10*a(n-10) + 2*a(n-12) + 4*a(n-13) + 4*a(n-14) - 4*a(n-15) - a(n-16) - 2*a(n-17) + a(n-18) + 2*a(n-19) - a(n-20) for n>19. - _Colin Barker_, Apr 07 2019
%t CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))^2,{x,0,50}],x] (* _Harvey P. Dale_, Jul 22 2012 *)
%o (Magma) n:=4; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H);
%o (Maxima) a(n):=floor(2*floor(-n/3)*cos(2*%pi*(n+1)/3)/81+(n+2)*cos(%pi*n/ 2)/128+(n+1)*(2835*(n^2+29*n+246)*(-1)^n+6*n^6+414*n^5+11556*n^4 +166944*n^3 +1320045*n^2+5489625*n+10008110)/ 17418240+1/2); /* _Tani Akinari_, Nov 14 2012 */
%o (PARI) Vec(1 / ((1 - x)^8*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40)) \\ _Colin Barker_, Apr 07 2019
%Y Column four of table A115994.
%Y Cf. A001400, A001399, A117485, A117487.
%Y Cf. A000027, A006918, A115994.
%K nonn,easy
%O 0,2
%A _Alford Arnold_, Mar 22 2006
%E Entry revised by _N. J. A. Sloane_, Mar 10 2007
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